doPhysics · Optics
Refraction, Total Internal Reflection and Prisms
Light slows down in glass and water and bends at their surfaces. Snell's law, apparent depth and the glass slab, total internal reflection, optical fibres and diamonds, mirages and rays in graded media, the prism and its minimum deviation, dispersion and the rainbow.
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Why this chapter
Why this chapter exists
A straw looks broken in a glass of nimbu-pani. A coin rises into view as a bucket fills, and a dry road in Rajasthan seems to hold water. All three happen because light changes speed when it passes from one medium into another, and so changes direction. This chapter follows light across flat surfaces: into glass and water, through slabs and layers, and through prisms. Past a certain angle it cannot get out at all, and that trap carries the internet in optical fibres and makes a diamond sparkle. Finally a prism, and a raindrop, spread white light into its colours.
You will learn
By the end of this chapter you will
- Use \(\mu = c/v\) and explain why light keeps its frequency but changes its wavelength in a medium.
- Apply Snell's law \(\mu_1\sin i = \mu_2\sin r\), and the relative refractive index, to rays at any surface.
- Find the apparent depth of an object, through one medium or several layers, and the speed of its image.
- Find the normal shift and the lateral shift produced by a glass slab.
- State the conditions for total internal reflection, find the critical angle, and solve problems on the circle of light, fibres and reflecting prisms.
- Use \(\mu\sin\theta = \text{constant}\) for layered and graded media, explain a mirage, and derive the path \(y = x^2/4a\).
- Use \(r_1 + r_2 = A\) and \(\delta = i + e - A\) for a prism, and find \(\mu\) from the minimum deviation.
- Decide when a ray cannot emerge from a prism.
- Use \(\delta = (\mu - 1)A\) for a thin prism, find angular dispersion and dispersive power, and design prism pairs that cancel deviation or dispersion.
- Explain the rainbow from refraction and reflection in raindrops.
Before you begin
Before you begin
From the chapter on Reflection, you should know the laws of reflection, the normal to a surface, and how a plane mirror forms an image. From the chapters on waves, you should know \(v = f\lambda\) and that a wave keeps its frequency when it enters a new medium. You should be comfortable with sine, cosine and tangent, small-angle approximations, and one simple integration.
Lessons
1
Refraction and Snell's Law
Light slows down in glass and water, so its wavelength shrinks while its frequency stays fixed. The refractive index \(\mu = c/v\), Snell's law \(\mu_1\sin i = \mu_2\sin r\), bending towards or away from the normal, and the relative refractive index.
Light slows in a mediumSnells lawRelative refractive index
2
Apparent Depth and the Glass Slab
Why water looks shallower than it is. Apparent depth \(h/\mu\) for near-normal viewing, layers that add their shifts, an image that moves slower than its object, the normal shift \(t(1 - 1/\mu)\) of a slab, and the lateral shift of a ray through it.
The coin at the bottom of a bucketShifts add through layersNormal shift through a slabLateral shift through a slab
3
Total Internal Reflection
Light going from a denser to a rarer medium bends away from the normal until, at the critical angle, it grazes the surface. Beyond it all the light is reflected. The circle of light above a lamp under water, the fish's window, optical fibres, reflecting prisms and the sparkle of a diamond.
The critical angleThe circle of light above a lampFibres prisms and diamonds
4
"Bending in Layers: Mirages and Graded Media"
Through flat layers \(\mu\sin\theta\) stays the same, so a ray climbing into thinner and thinner layers bends over and turns back. The mirage on a hot road, and the curved path of a ray in a medium whose refractive index changes smoothly, \(y = x^2/4a\).
Mu sin theta is constant across layersThe mirage on a rajasthan roadA curved ray in a graded medium
5
The Prism
A ray through a prism bends twice towards the base. The relations \(r_1 + r_2 = A\) and \(\delta = i + e - A\), the minimum deviation at the symmetric path and the formula for \(\mu\), grazing emergence and prisms that let no ray through, and the thin prism \(\delta = (\mu - 1)A\).
Refraction through a prismMinimum deviationGrazing emergence and no emergenceThe thin prism
6
Dispersion and the Rainbow
The refractive index of glass is a little larger for violet than for red, so a prism spreads white light into colours. Angular dispersion and dispersive power, prism pairs that cancel the deviation or the dispersion, and the rainbow, made by refraction and reflection in raindrops.
White light splits into coloursDispersive powerDispersion without deviation and the reverseThe rainbow
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- achromatic combination
- Two prisms or lenses of different glasses arranged so that their dispersions cancel, while some deviation remains.
- angle of deviation
- The angle \(\delta\) between the incident ray, continued, and the emergent ray. For a prism, \(\delta = i + e - A\).
- angle of emergence
- The angle between the ray leaving a prism or a slab and the normal to the face through which it leaves.
- angle of incidence
- The angle between an incident ray and the normal to the surface at the point where the ray arrives.
- angle of refraction
- The angle between a refracted ray and the normal to the surface at the point where the ray crosses it.
- angle of the prism
- The angle \(A\) between the two refracting faces of a prism.
- angular dispersion
- The angle between the violet and the red rays leaving a prism, \(\delta_v - \delta_r\). For a thin prism it is \((\mu_v - \mu_r)A\).
- apparent depth
- The depth at which an object in a denser medium appears when seen from a rarer one. Seen near the normal from air, it is the real depth divided by \(\mu\).
- cladding
- The layer of lower refractive index around the core of an optical fibre, at whose boundary the light is totally reflected.
- critical angle
- The angle of incidence in the denser medium for which the angle of refraction is \(90°\). It is given by \(\sin C = \mu_r/\mu_d\).
- crown glass
- A common optical glass of low dispersion, with \(\mu\) about 1.5 and \(\omega\) about 0.08 in this chapter's problems.
- deviation
- The angle through which a ray is turned from its original direction by a surface, a slab, a prism or a drop.
- dispersion
- The splitting of white light into its colours, because the refractive index of a medium is different for different wavelengths.
- dispersive power
- The angular dispersion divided by the mean deviation, \(\omega = (\mu_v - \mu_r)/(\mu_y - 1)\). It is a property of the material alone.
- flint glass
- An optical glass containing lead, with a higher refractive index and dispersion than crown glass, \(\mu\) about 1.6 and \(\omega\) about 0.1 here.
- graded medium
- A medium whose refractive index changes smoothly from place to place, such as air heated from below. A ray in it curves towards the higher refractive index.
- grazing emergence
- Emergence at \(90°\), with the ray leaving along the surface. It happens when the ray meets that face from inside at exactly the critical angle.
- grazing incidence
- Incidence at \(90°\), with the ray skimming along the surface as it arrives.
- lateral shift
- The perpendicular distance between a ray entering a parallel-sided slab and the parallel ray that leaves it, \(t\sin(i - r)/\cos r\).
- minimum deviation
- The least deviation \(\delta_m\) a prism can give, reached when the ray passes symmetrically, with \(i = e\).
- normal
- The line at right angles to a surface at the point where a ray meets it. Angles of incidence, reflection and refraction are measured from it.
- normal shift
- The distance by which an object seen through a medium or a slab appears moved towards the eye, along the normal. For a slab it is \(t(1 - 1/\mu)\).
- optical fibre
- A thin glass thread whose core has a higher refractive index than the cladding around it. It guides light along its length by total internal reflection.
- optical path
- The distance \(\mu d\) that light would travel in vacuum while it crosses a thickness \(d\) of a medium of refractive index \(\mu\).
- optically denser
- Of two media, the one with the larger refractive index, in which light travels more slowly. It need not have the larger mass density.
- principle of reversibility
- If the direction of a ray of light is reversed, it retraces its path exactly, through every reflection and refraction.
- prism
- A transparent block with two flat refracting faces inclined to each other. A ray through it is deviated towards the base.
- refraction
- The change in the direction of light as it passes at a slant from one medium into another, caused by the change in its speed.
- refractive index
- The ratio \(\mu = c/v\) of the speed of light in vacuum to its speed in a medium. Glass has about 1.5, water \(\tfrac43\). Other books write \(n\).
- relative refractive index
- The ratio \({}_1\mu_2 = \mu_2/\mu_1 = v_1/v_2\) for light passing from medium 1 into medium 2. It equals \(\sin i/\sin r\) at their boundary.
- spectrum
- The band of colours, from red to violet, into which white light is spread by dispersion.
- thin prism
- A prism whose angle is only a few degrees. For rays near normal incidence it deviates every ray by \((\mu - 1)A\).
- total internal reflection
- The reflection of all the light at a boundary. It happens when light going from a denser to a rarer medium meets the boundary at more than the critical angle.